If a category theoretic proof is what you're after, you'll first want a universal property which characterizes intersections and sums of ideals. Honestly, it's not worth the effort. Just chase through what intersection and sum mean - it will probably be a lot easier.

A polynomial [imath]m(x)[/imath] is the minimal polynomial for a number [imath]\alpha[/imath] exactly when [imath]\alpha[/imath] is a root and [imath]m(x)[/imath] is irreducible over the relevant field (in our case, [imath]\mathbf{Q}[/imath]).

Remember: [imath]a^xb^x=(ab)^x[/imath]. That includes cases where [imath]x=1/3[/imath] (like you have). With this in mind, just multiply two generic elements and see that the product can be massaged into the desired form.

Operating systems all suck, so I just use a free one . lolwut? Ubuntu, while free, is still an operating system. That said, I'll echo the suggestion to use Ubuntu. As mentioned, it's free and the distro is fairly newbie friendly. Debian which you mentioned is what Ubuntu is based off of, but it's n...

Exactly. \sin(\pi/4)=\sin(3\pi/4)=\sin(9\pi/4)=\ldots=\sqrt{2}/2 , but for the inverse to be a function, you can't have \arcsin(\sqrt(2)/2)=\pi/4, 3\pi/4\ldots. , so we take the principle value, which for arcsin, will give you whichever value works in the inte...

G/H , not G\backslash H . G\backslash H would be the group resulting from their set difference or something (if it's even a group). Using the first isomorphism theorem suggests to me finding a homomorphism \phi such that \ker \phi = H . Then G/H could be 'identified' as the image of \phi .

I also am trying to make a decent proof that e^iπ=-1. I started with an assumption of Euler's law (not sure if that's the right name but it was one of Euler's) (e^ix=cosx+isinx) and am currently in the process of proving that. I've got to prove that the taylor series give e^x. Once I've done that I...

The "multiplication" you're talking about is the dot product. The dot product is done by multiplying together the corresponding components and then adding all these components together. For example, \begin{pmatrix} 1\\2\\3\end{pmatrix} \cdot \begin{pmatrix}4\\-5\\6\end{pmatrix} = (1...

Random brain block or something... Can we combine a proof by (strong) induction with a proof by contradiction? That is, can we prove a base case, make an inductive hypothesis, and then contradict the inductive hypothesis? Ah hell, a sketch of my proof might help. I'm proving that successive Fibonacc...

Most hex editors I've used don't allow insertion or deletion at all. (It's too easy to do by accident and the file formats people typically use hex editor on aren't safe to shift things around in). Maybe start with just allowing users to overwrite characters without moving them? Well implementing t...

So I've been working on my hex editor lately, and I now have a functioning gui implemented. Yay! My next course of action is to allow the user to actually modify the files, but I'm not sure how to best implement this... Inserting or deleting in place will be hugely inefficient since the file is just...

Hey all I have some header files that look something like this //file1.h typedef struct blah1_t { // ... blah2 something; } blah1; // file2.h typedef struct blah2_t { // ... (only built in types) } blah2; void function(blah1 *); It won't compile no matter which files I include in each, and if I try ...

In C, the length of an array declared on the stack (which is what all the variables in your code above are) must be known at compilation time. That means you can't do what you want where you ask the user for the length of the array, and then declare it to be that length. To get around this, you shou...

I suppose I could just code up a quick test program to find out, but does anyone know what happens if you call ftell() on a file when the current location exceeds the size of a long (the return type)? Does it return a -1, or does it overflow?

What project(s) do you currently have on the go? I'm currently working on an ncurses hex editor in C. I've never used the ncurses library before, and documentation for it isn't fantastic making it a bit difficult, but so far I seem to be getting along OK. I just finished the basics of a GUI, so now ...

I figure there are some interesting qualities of palindromic numbers. In all honesty, no, there aren't. The property of being a palindrome isn't relevant to any "serious" mathematics. Te elaborate a bit, which numbers are palindromic is entirely dependent upon trivial issues such as the b...

Lord of the Rings - Despite finding the premise of the book legitimately interesting, the plot just progresses so damn slow that I lose interest. I did enjoy The Hobbit though. Neuromancer - Again, the idea of the book is good, but the writing style bugs me. I may give it another go sometime. Great ...